Báo cáo toán học: " Tight Quotients and Double Quotients in the Bruhat Order" doc

Inthis paper, we 1 provide a general characterization of tightness for one-sidedquotients, 2 classify all tight one-sided quotients of finite Coxeter groups, and3 classify all tight doub

Tight Quotients and Double Quotients in the Bruhat Order

John R Stembridge*

Department of MathematicsUniversity of MichiganAnn Arbor, Michigan 48109–1109 USA

jrs@umich.edu

Dedicated to Richard Stanley on the occasion of his 60th birthday

Submitted: Aug 17, 2004; Accepted: Jan 31, 2005; Published: Feb 14, 2005

Mathematics Subject Classifications: 06A07, 20F55

Abstract

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxetergroup is the conjunction of its projections onto quotients by maximal parabolicsubgroups Similarly, the Bruhat order is also the conjunction of a larger number

of simpler quotients obtained by projecting onto two-sided (i.e., “double”) tients by pairs of maximal parabolic subgroups Each one-sided quotient may berepresented as an orbit in the reflection representation, and each double quotientcorresponds to the portion of an orbit on the positive side of certain hyperplanes

quo-In some cases, these orbit representations are “tight” in the sense that the rootsystem induces an ordering on the orbit that yields effective coordinates for theBruhat order, and hence also provides upper bounds for the order dimension Inthis paper, we (1) provide a general characterization of tightness for one-sidedquotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and(3) classify all tight double quotients of affine Weyl groups

0 Introduction.

The Bruhat orderings of Coxeter groups and their parabolic quotients have a longhistory that originates with the fact that these posets (in the case of finite Weyl groups)record the inclusion of cell closures in generalized flag varieties

Some of the significant early papers on the combinatorial aspects of this subject

in-clude the 1977 paper of Deodhar [D1] providing various characterizations of the Bruhat order (including some that will be essential in this work), the 1980 paper of Stanley [St]

in which Bruhat orderings of finite Weyl groups and their parabolic quotients are shown

to be strongly Sperner, and the 1982 paper of Bj¨orner and Wachs in which the Bruhat

order is shown to be lexicographically shellable [BW].

* This work was supported by NSF grant DMS–0245385.

In this paper, we investigate the explicit assignment of coordinates for the Bruhatorder By a “coordinate assignment” for a poset P , we mean an order-embedding

P → R d ; i.e., an injective map f : P → R d such that x < y in P if and only if

f (x) < f (y) in the usual (coordinate-wise) partial ordering of R d The minimum such

d for which this is possible is known as the order dimension of P , and denoted dim P

For example, Proctor [P1] has given coordinates for the Bruhat orderings of the classical finite Coxeter groups and their quotients, and more recently, Reading [R] has

determined the exact order dimensions of the Bruhat orderings of An, Bn, H3, and H4.

It would be interesting to have a uniform construction of coordinates for the Bruhatorders of finite Weyl groups, perhaps based directly on the geometry of flag varieties as

in Proposition 7.1 of [P1] for type A For the infinite Coxeter groups, perhaps the most

interesting question is the classification of those groups for which the Bruhat ordering

is finite-dimensional Indeed, Reading and Waugh [RW] have shown that there are

Coxeter groups whose Bruhat order has infinite order dimension, and infinite Coxeter

groups (such as the affine Weyl groups of type A) with finite order dimension.

Our initial motivation for this work began with the observation that for each finite

Weyl group W and associated affine Weyl group f W , the two-sided (parabolic) quotient

W \f W /W may be naturally identified with the dominant part of the co-root lattice.

We were surprised to realize that the Bruhat ordering of W \f W /W is isomorphic to the

usual ordering of dominant co-weights: moving up in this Bruhat order is equivalent toadding positive combinations of positive co-roots (Later, we learned from M Dyer that

this is mentioned explicitly in Section 2 of [L].) This meant that the various remarkable properties of the partial order of dominant (co-)weights (see for example [S2]) could be

transfered to the Bruhat ordering of certain two-sided quotients of affine Weyl groups

At this point, we began to investigate more general instances of this phenomenon.Indeed, it is always possible to identify a one-sided parabolic quotient of any Coxetergroup with the orbit of a point in the reflection representation, and a two-sided (or

“double”) quotient corresponds to the part of an orbit on the positive side of certainhyperplanes In these terms, a necessary condition for moving up in the Bruhat orderrequires adding (or subtracting, depending on conventions) positive combinations of pos-itive roots The interesting question is one of identifying when this necessary condition

is sufficient That is, when do the root coordinates of an orbit, or the portion sponding to some double quotient, provide an order embedding of the correspondingBruhat order? The main goal of this paper is to identify these “tight” quotients

corre-An outline of the paper follows

In Section 1, we discuss the details of using the reflection representation of a Coxetergroup to model the Bruhat orderings of its parabolic quotients We also review a

key result of Deodhar (see Theorem 1.3) that allows the Bruhat ordering of W to be

recovered from its projections onto one-sided or two-sided quotients

In Section 2, we formalize the notion of a tight quotient, and prove a purely theoretic characterization of the tight one-sided quotients (Theorem 2.3): the Bruhat

order-ordering of W/WJ is tight if and only if the Bruhat ordering of WI \W/W J is a chain

for every maximal parabolic subgroup WI of W We also point out that the Bruhat

orderings of minuscule (one-sided) quotients are always tight.

In Section 3, we classify the tight one-sided quotients of finite Coxeter groups Weexpected the results to include only a few instances beyond the minuscule cases (afrequent outcome in the theory of finite Coxeter groups), but were instead surprised

to discover that there are many other examples, including quotients by non-maximalparabolic subgroups

In the course of deriving the classification, we develop two significant necessary ditions for tightness The first involves the “stratification” of an orbit relative to theaction of a parabolic subgroup, and the second involves confining a face of the dominantchamber inside a face of the “double weight arrangement” of hyperplanes (an arrange-ment that is in general much larger than the usual arrangement defined by the roothyperplanes) In fact, both of these necessary conditions may be used to provide char-acterizations of tightness (see Lemma 3.3, Theorem 3.9, and Corollary 3.10), although

con-our proofs of the latter two depend a posteriori on the classification.

In the final two sections, we focus on the affine Weyl groups For these groups,there are two natural representations: the first is the usual reflection representation—available for all Coxeter groups—in which the group is represented via linear operators;

in the second, one uses affine transformations In Section 4, we present a dictionaryfor translating between these two points of view, and prove that there are no one-sided

or double quotients that are tight relative to the reflection representation, apart fromsome trivial cases (Theorem 4.9) In contrast, we show that double quotients with bothfactors of minuscule type are tight relative to the affine representation (Theorem 4.10)

In Section 5, we turn to the classification of quotients of affine Weyl groups that aretight relative to the affine representation In particular, Theorem 5.12 and Corollary 5.13provide a classification of all double quotients with a tight embedding in some affineorbit; we find that the left factor must be of minuscule type, but there is a largernumber of possibilities for the right factor The proof has a structure similar to theone in Section 3—we find that there are affine analogues of orbit stratification and thedouble weight arrangement that provide characterizations of tightness similar to those

we develop for finite Coxeter groups (see Theorems 5.10 and 5.11)

Acknowledgment.

I would like to thank Nathan Reading for many helpful discussions

1 The Bruhat order.

Let (W, S) be a Coxeter system Via the reflection representation, one may view W

as a group of isometries of some real vector space V equipped with a (not necessarily

positive definite) inner producth , i In particular, we may associate with W a

centrally-symmetric, W -invariant subset Φ ⊂ V − {0} (the root system) so that the reflections

in W are the linear transformations sβ : λ 7→ λ − hλ, βiβ ∨ , where β varies over Φ, and β ∨ := 2β/hβ, βi denotes the co-root corresponding to β.1 In this framework, the

1For the details of this construction, we refer the reader to (for example) Chapter 5 of [H], although

it should be noted that the normalizationhβ, βi = 1 for β ∈ Φ in [H] may be relaxed—rescaling each

W -orbit of roots by an arbitrary positive scalar has no significant effect on the general theory.

generating set S is the set of simple reflections: for each s ∈ S one may choose a root

α (designated to be simple) so that s = s α, and these choices may be arranged so thatevery root is in either the nonnegative or nonpositive span of the simple roots Thus Φ

is the disjoint union of Φ+ (the positive roots) and Φ− =−Φ+ (the negative roots).

For w ∈ W , let `(w) denote the minimum length of an expression w = s1· · · s l (si ∈ S) A key relationship between the root system and length is the fact that

`(w) < `(s β w) ⇔ w −1 β ∈ Φ+ (w ∈ W, β ∈ Φ+), (1.1) and the Bruhat ordering of W may be defined as the transitive closure of the relations

w <B s β w

for all w ∈ W and β ∈ Φ+ satisfying either of the equivalent conditions in (1.1)

For each J ⊂ S, we let WJ denote the parabolic subgroup of W generated by J, and

ΦJ ⊂ Φ the corresponding root subsystem One knows that

W J : ={w ∈ W : `(ws) > `(w) for all s ∈ J},

J W : = {w ∈ W : `(sw) > `(w) for all s ∈ J}

are the unique sets of coset representatives for W/WJ and WJ \W (respectively) that

minimize length, and similarly (Exercise IV.1.3 of [B])

I W J :=I W ∩ W J

is the unique set of length-minimizing representatives for the double cosets WI \W/W J

A Orbits and one-sided quotients.

If θ ∈ V is dominant (i.e., hθ, βi > 0 for all β ∈ Φ+), then the W -stabilizer of θ is the parabolic subgroup WJ , where J = {sα ∈ S : hθ, αi = 0} This allows W/W J to be

identified with the W -orbit of θ, and as previously noted in [S4], the following result

shows that the poset structure of W J (as a subposet of (W, <B)) may be transported

to a partial ordering on W θ by taking the transitive closure of the relations

µ <Bs β(µ) for all β ∈ Φ+ such that hµ, βi > 0.

Proposition 1.1 [S4] Assume θ ∈ V is dominant with stabilizer WJ

(a) Evaluation (i.e., w 7→ wθ) is an order-preserving map (W, <B)→ (W θ, <B) (b) The evaluation map restricts to a poset isomorphism (W J , <B)→ (W θ, <B).

Proof (a) If w <B s β w is a covering relation in (W, <B), then (1.1) implies that w −1 β

is a positive root, so hwθ, βi = hθ, w −1 βi > 0 Hence either wθ = s β wθ (if hwθ, βi = 0)

or wθ <B s β wθ (if hwθ, βi > 0), so wθ 6B s β wθ in both cases.

(b) Since WJ is the stabilizer of θ, it is clear that the evaluation map is a bijection between W J and W θ, so we need only to show that the inverse map is order-preserving Thus suppose we have a covering relation µ <B s β (µ) in (W θ, <B) for some root β ∈ Φ+.

We necessarily have hµ, βi > 0, so if w is the unique member of W J such that µ = wθ,

then hθ, w −1 βi = hµ, βi > 0, so w −1 β is a positive root and w <B s β w.

Now let x ∈ WJ be the unique element such that sβ wx ∈ W J It follows easily

from the definition that each member of W J is the Bruhat-minimum of its coset, so

w 6B wx Furthermore, it is clear that s β wx and wx must be related in Bruhat order.

However, sβ wx <B wx would contradict (a), so in fact w 6B wx <B s β wx and the

result follows 

Remark 1.2 (a) One complication for infinite Coxeter groups is that the bilinearform h , i may be degenerate on V However, it is always possible to replace V with a

larger space and extend the bilinear form in a non-degenerate way This allows us to

identify V with its dual space, and guarantee that for every parabolic subgroup WJ,

there is a dominant point in V whose stabilizer is WJ

(b) (Proposition 3 of [P1]) The quantityhµ+tβ, µ+tβi is a quadratic function of t,

and µ 7→ hµ, µi is constant on W -orbits, so for each root β there is at most one other point in the W -orbit of µ of the form µ + tβ (namely, sβ(µ)) It follows that the Bruhat ordering of W θ may alternatively be defined as the transitive closure of all relations

µ < ν (µ, ν ∈ W θ) such that µ − ν is a positive multiple of a positive root.

(c) One knows that the Bruhat ordering of W J is graded by length (e.g., see [D1]).

If we transport this to (W θ, <B), we obtain the rank function

r(µ) := |{β ∈ Φ+ :hµ, βi < 0}| (µ ∈ W θ).

Indeed, given µ = wθ and w ∈ W J , there are three possibilities for each β ∈ Φ+,depending on the sign of hµ, βi = hθ, w −1 βi: if it is negative, then w −1 β ∈ Φ −; if it

is positive, then w −1 β ∈ Φ+; if it vanishes, then w −1 β ∈ Φ J , and hence w −1 β ∈ Φ+

(otherwise, we contradict (1.1) and the fact that w ∈ W J ) Hence r(µ) = |Φ+∩ wΦ − |,

a well-known expression for the length of w (e.g., see Section 5.6 of [H]).

Let πJ : W → W J denote the natural projection map (i.e., πJ (xy) = x for all x ∈ W J and y ∈ WJ ) An immediate corollary of Proposition 1.1 is the well-known fact that πJ

is order-preserving As a sort of converse to this, we have

Theorem 1.3 (Deodhar [D1]) For all I, J ⊆ S and x, y ∈ W , we have

π I∩J (x) 6Bπ I∩J (y) if and only if πI (x) 6B π I (y) and πJ (x) 6Bπ J (y).

It will be convenient for what follows to use the abbreviation hsi for S − {s}.

Corollary 1.4 For all J ⊆ S and all x, y ∈ W J , we have

x 6By if and only if π hsi(x) 6B π hsi(y) for all s ∈ S − J.

It follows that an assignment of coordinates for the Bruhat ordering of any parabolic

quotient W J (including the full group W in the case J = ∅) may be produced once we

have coordinates for the quotients by maximal parabolic subgroups In particular, wemay deduce bounds on the order dimension; viz.,

dim(W J , <B) 6 X

s∈S−J dim(W hsi , <B).

B Double quotients.

Given I ⊆ S and a dominant θ ∈ V with stabilizer WJ, let

(W θ)I :=

µ ∈ W θ : hµ, αi > 0 for all α ∈ Φ+I

denote the subset of W θ that is dominant with respect to ΦI The following result

shows that the subposet of (W θ, <B) formed by (W θ)I is isomorphic to the Bruhatordering of the double quotient I W J, and that this subposet may be generated by theapplication of certain reflections

Proposition 1.5 Assume θ ∈ V is dominant with stabilizer WJ

(a) If w ∈ I W , then wθ ∈ (W θ) I

(b) If w ∈ W J and wθ ∈ (W θ) I , then w ∈ I W J (Hence, the evaluation map

w 7→ wθ restricts to a bijection between I W J and (W θ) I )

(c) The partial ordering of (W θ)I , as a subposet of (W θ, <B), is generated by the

transitive closure of the relations µ <B s β(µ) for all µ ∈ (W θ)I and β ∈ Φ+such that s β (µ) ∈ (W θ)I and hµ, βi > 0.

Proof (a) If w ∈ I W , then we have w −1 α ∈ Φ+ for all simple α ∈ ΦI, and hence also

for all α ∈ Φ+I It follows thathwθ, αi = hθ, w −1 αi > 0 for all such α; i.e., wθ ∈ (W θ) I

(b) Suppose w ∈ W J and wθ ∈ (W θ)I If w failed to be in I W , then we would have

`(s α w) < `(w) for some simple α ∈ Φ I , whence w −1 α ∈ Φ − andhwθ, αi 6 0 However,

given that wθ is ΦI-dominant, this is possible only ifhwθ, αi = 0, and hence s α wθ = wθ,

contradicting the fact that w is the shortest element of W that maps θ to wθ Thus

w ∈ I W ∩ W J =I W J

(c) It suffices to show that every relation µ <B ν involving elements µ, ν ∈ (W θ) I is

a transitive consequence of the given relations For this, let x, y ∈ W be the shortest elements such that xθ = µ and yθ = ν; we then have x, y ∈ W J and (by Proposition 1.1)

x <B y, so there is a maximal chain x = x0 <B x1 <B · · · <B x l = y in (W, <B)

However, we also have x, y ∈ I W from (b), and recall that ( I W, <B) and (W J , <B)are both graded by the length function (cf Remark 1.2(c)), so we may assume that

the maximal chain from x to y is chosen so that each xi is in I W Now since covering

relations in (W, <B) are generated by reflections, it follows that the image of this chain

under the map w 7→ wθ is a chain of the desired form, by (a) 

Remark 1.6 (a) The above result shows that (W θ)I (and hence indirectly, thedouble quotient I W J ) may be obtained by generating the smallest subset of V that

Figure 1: The Bruhat ordering of a double quotient of S4.

contains θ and is closed under the operation of applying a reflection sβ (β ∈ Φ) whenever

the result stays within the ΦI-dominant chamber In general, this is vastly more efficient

than traversing the full W -orbit of θ and selecting those points that are ΦI-dominant

(b) Given x, y ∈ I W J such that x <B y, there need not exist a maximal chain in

(W, <B) from x to y such that each intermediate term is also in I W J Otherwise,(I W J , <B) would have to be graded by length, contrary to the example in Figure 1

Example 1.7 (a) Taking W to be the symmetric group Sn, acting on V = R n by

permuting coordinates, the Bruhat ordering of an orbit W θ is generated by tions that increase the number of inversions For a given choice of I, the ΦI-dominant

transposi-part of W θ consists of those permutations of θ that have do not have a strict descent

at certain fixed positions (depending on I) For example, if n = 4 and θ = (0, 1, 1, 2), there are 7 permutations of θ that do not have a descent between the first and second

positions, and the Bruhat ordering of these 7 points is illustrated in Figure 1

(b) Another way to index double cosets WI \W/W J in the symmetric group case is

to use contingency tables; i.e., nonnegative integer matrices with fixed row and column

sums depending on I and J [DG] Moving to a higher table in the Bruhat ordering

corresponds to adding [−1 1

1 −1] to some 2× 2 submatrix of a given contingency table.

Returning to the general case, let λI : W → I W denote the left sided analogue of

the projection πI ; thus, λI (yx) = x for all y ∈ WI and x ∈ I W By abuse of notation,

we will also use λI to denote a map on V in which λI (µ) is defined to be the unique member of the WI -orbit of µ that is dominant with respect to ΦI Proposition 1.5(a)

shows that these two uses are compatible with evaluation; i.e., λI (w)θ = λI (wθ) for all dominant θ and all w ∈ W

If we apply left projections to right quotients (a suggestion of Reading), we obtain

Proposition 1.8 Let I, J ⊆ S and assume θ ∈ V is dominant with stabilizer WJ

(a) We have λI (W J) =I W J

(b) The map λI : (W θ, <B)→ ((W θ) I , <B) is order-preserving.

(c) For all µ, ν ∈ W θ, we have µ 6Bν if and only if λ hsi(µ) 6Bλ hsi(ν) for all s ∈ S.

Proof (a) Toward a contradiction, suppose we have w ∈ W J and λI (w) / ∈ W J.

Among all such counterexamples, choose w so as to minimize length We must have

w / ∈ I W ; otherwise, λ I (w) = w ∈ W J , and w would fail to be a counterexample Thus

`(sw) < `(w) for some s ∈ I It follows also that sw / ∈ W J ; otherwise, sw would be

a shorter counterexample Hence there is some t ∈ J such that `(swt) < `(sw), and therefore `(wt) 6 `(swt) + 1 = `(sw) < `(w), contradicting the fact that w ∈ W J

(b) Given µ, ν ∈ W θ, there exist unique x, y ∈ W J such that xθ = µ and yθ = ν Since w 7→ w −1 is an automorphism of (W, <B) that interchanges W I and I W , and

right projections are order preserving (Proposition 1.1), it follows that λI : W → I W is

also order-preserving Hence,

µ 6Bν ⇒ x 6B y ⇒ λ I (x) 6Bλ I (y) ⇒ λI (µ) = λI (x)θ 6B λ I (y)θ = λI (ν),

the first and third implications being a consequence of Proposition 1.1

(c) Given µ, ν ∈ W θ and x, y ∈ W J as above, (a) implies λhsi(x), λhsi(y) ∈ W J, so

λ hsi(µ) 6Bλ hsi(ν) ⇒ λhsi(x) 6B λ hsi (y)

by Proposition 1.1 Again using the fact that w 7→ w −1 is an order automorphism,

it follows that there is a left-handed version of Deodhar’s criterion (Corollary 1.4) In

particular, the above implications for all s ∈ S combine to imply that x 6B y, and

hence µ 6B ν by evaluation The converse implication follows from (b) 

The above results show that (W J , <B) is the conjunction of its left projections ontothe double quotients (hsi W J , <B) More generally, the Bruhat ordering of W or any

of its one-sided or double quotients is the conjunction of its projections onto maximaldouble quotients, and this implies a bound on the order dimension; viz.,

For example, in the symmetric group case, it is easy to show that the Bruhat ordering

of each maximal double quotient is a chain; hence the above bounds immediately yield

dim(Sn , <B)6 (n − 1)2.

Reading has shown that the order dimension of (Sn , <B) is bn2/4c (see [R]).

2 Tight quotients.

Having represented the Bruhat orderings of the one-sided and double quotients of W

on the W -orbits of various (dominant) points θ in a real vector space V , it is natural to

investigate the extent to which these representations may be used provide a coordinateembedding of the corresponding posets

Recall that if µ <B ν is a covering relation in (W θ, <B), then µ − ν is a positive

multiple of a positive root, and thus in the (simplicial) cone R+Φ+ generated by the

positive roots We define the standard (or root) ordering of V to be the partial order

µ < ν if µ − ν ∈ R+Φ+.

In these terms, the Bruhat order is consistent with the dual of the standard order; i.e.,

If this is an order embedding (i.e., µ 6B ν ⇔ µ < ν for all µ, ν ∈ W θ), then we say that

the Bruhat ordering of W θ is tight, or simply that (W θ, <B) is tight More generally,

any subposet of (W θ, <B) with this property is also said to be tight In particular, the

orbit representation of the Bruhat ordering of some double quotient, say ((W θ)I , <B),

is tight if µ 6B ν ⇔ µ < ν for all µ, ν ∈ (W θ) I

For example, consider the double quotient of S4 discussed in Example 1.7(a) and

illustrated in Figure 1 The pair µ = (0, 2, 1, 1) and ν = (1, 1, 2, 0) are incomparable in the Bruhat order, and yet µ − ν = (−1, 1, −1, 1) is the sum of two simple roots, so we have µ  ν This representation of the double quotient is therefore not tight.

Remark 2.1 (a) The simple roots generate the cone R+Φ+, so if X ⊆ W θ is tight, then the coordinates of θ − µ with respect to the simple roots, as µ varies over X, provide a coordinate embedding of the Bruhat ordering of X In particular, the order

dimension of any tight subposet of the Bruhat order is at most |S|.

(b) If we renormalize the root system Φ, independently replacing each W -orbit of roots by some positive scalar multiple, then the dominant chamber, the set of W - orbits in V , the cone spanned by the positive roots, and the standard ordering are all

unchanged Thus, tightness does not depend on how the root system is normalized

Example 2.2 It is implicit in the work of Proctor (see Proposition 4.1 of [P2]) that

if W is finite and θ is minuscule (i.e., hθ, β ∨ i ∈ {0, ±1} for all roots β), then the Bruhat

ordering of W θ is tight Indeed, if θ is minuscule, then the same is true for every point

in the W -orbit of θ, and thus every reflection acts on W θ by adding (or subtracting) a root It follows that if µ  ν in W θ, then µ − ν = P

c i α i for certain positive integers

c i and simple roots αi Furthermore, given that W is finite, it is necessarily the case

that h , i is positive definite on Span Φ, and hence hµ − ν, α ∨

i i > 0 for some i, whence

hµ, α ∨

i i = 1 or hν, α ∨

i i = −1 Thus we obtain µ <Bµ − α i < ν or µ < ν + αi <Bν and

it follows by induction with respect to P

c i that µ <Bν.

Theorem 2.3 The Bruhat ordering of W θ is tight if and only if for all s ∈ S, the

Bruhat ordering of the double quotient (W θ) hsi is a chain.

Our proof will rely on the following pair of lemmas The first of these may be known, but we have not seen it in the literature

well-Lemma 2.4 If Φ is infinite and irreducible, then for every µ ∈ Span Φ, there exist

roots γ such that γ  µ.

Proof Consider the collection of subsets J ⊆ S such that the claimed property is

true for all µ ∈ Span ΦJ There must be nonempty sets J with this property; otherwise,

the coordinates of the positive roots would be bounded, and hence Φ would have an

accumulation point In turn, this would force the set of reflections in W to have an accumulation point in the usual topology of GL(V ), contradicting the fact that Coxeter

groups are discrete (e.g., see Section 6.2 of [H]).

Now suppose that J is maximal with respect to the above property Given that Φ

is irreducible (and assuming J 6= S), there must exist simple reflections sα ∈ J and

s β ∈ J such that the corresponding nodes of the Coxeter diagram are adjacent; i.e., /

hα, βi < 0 However since J is maximal, there must be some µ ∈ Span Φ J and a scalar

c > 0 such that there is no root γ  µ + cβ On the other hand, we have s α ∈ J, so

we can find roots γ  µ + tα for all t > 0 The coefficient of β in all such roots must

necessarily be 6 c, and since hα 0 , βi 6 0 for all simple roots α 0 6= β, it follows that

hγ, β ∨ i 6 2c + thα, β ∨ i By choosing t sufficiently large, we force hγ, β ∨ i < −c, whence

s β(γ)  γ + cβ  µ + cβ, contradicting the choice of µ and c 

Lemma 2.5 Assume W is irreducible If µ, ν ∈ V are dominant and µ − ν ∈ Span Φ,

then there exist points µ 0 in the W -orbit of µ such that ν < µ 0 , except possibly if W is infinite and acts trivially on µ.

Proof If Φ (and hence W ) is finite, then there is an anti-dominant member of the

W -orbit of µ, say µ 0 If ν < µ 0 failed, then there would be a nonempty set of simple

roots α with negative coefficients in ν − µ 0 Choosing β to be a (necessarily positive)

root that is dominant relative to the corresponding (finite) parabolic root subsystem

ΦJ, we would therefore have hα, βi 6 0 for simple roots α not in Φ J (since hα, α 0 i 6 0

for distinct simple roots) andhα, βi > 0 for simple roots α in Φ J, with at least one strictinequality among the latter cases It follows that hν − µ 0 , βi < 0, so either hν, βi < 0 or

hµ 0 , βi > 0, contradicting the fact that ν is dominant and µ 0 is anti-dominant

If Φ is infinite, then by Lemma 2.4 there is a positive root β such that ν < µ − β Also, given that W does not act trivially on µ, there is a (simple) root α such that

hµ, αi > 0 Replacing β with a higher root if necessary, we may therefore assume that

hµ, β ∨ i > 1, and hence ν < µ − β < s β (µ) 

We remark that the above lemma fails without the exception for trivial actions Forexample, if the bilinear formh , i is degenerate on the span of Φ, then there exist nonzero

W -fixed points δ ∈ Span Φ It cannot be the case that both δ < 0 and 0 < δ, so taking

either µ = δ, ν = 0 or µ = 0, ν = δ would produce a counterexample.

Proof of Theorem 2.3 ( ⇒) Suppose µ, ν ∈ (W θ) hsi are incomparable in the Bruhatorder Interchanging µ and ν if necessary, we may assume that the coefficient of α in

ν − µ is nonnegative, where α denotes the simple root corresponding to s We claim

that there is an element µ 0 in the Whsi-orbit of µ such that ν < µ 0 If this failed, then

by Lemma 2.5, there would have to be an infinite irreducible component WJ of Whsi

that acts trivially on µ, and a simple root α 0 ∈ Φ J such that the coefficient of α 0 in

ν − µ is negative Note that W J∪{s} must also be irreducible; otherwise, WJ would be

an irreducible component of W and the coefficient of α 0 would be constant on W θ Now since µ is dominant with respect to Φhsi but not Φ-dominant (otherwise µ = θ and we contradict the fact that µ and ν are incomparable), it must be the case that

hµ, αi < 0 We also have hµ, βi = 0 for β ∈ Φ J, so hµ, βi < 0 all positive roots β in

ΦJ∪{s} − Φ J However, there are infinitely many such roots (Lemma 2.4), contradicting

the fact that the number of these roots is bounded by the rank of µ (see Remark 1.2(c)) Given that (W θ, <B) is tight, we may conclude that ν 6B µ 0 , and hence ν 6B µ, since

the left projection map λhsi preserves the Bruhat order (Proposition 1.8(b)) However,

this contradicts the hypothesis that µ and ν are incomparable.

(⇐) Given µ, ν ∈ W θ such that µ < ν, the left projections λ hsi (µ) and λhsi(ν) must

be Bruhat-related for all s ∈ S, since each of ((W θ)hsi , <B) is a chain We cannot

have λhsi(ν) <B λ hsi(µ); otherwise, since λhsi(µ) is the Bruhat-minimum of its Whsi

-orbit, the first step in any Bruhat-decreasing chain from λhsi (µ) to λhsi(ν) would be a reflection not in Whsi, and thus the simple root α corresponding to s would appear in

λ hsi (µ) − λhsi(ν) with a coefficient < 0 However, the coefficient of α is constant on

W hsi-orbits, so this contradicts the fact that µ < ν Thus we have λhsi(µ) 6B λ hsi(ν) for all s ∈ S, and hence µ 6B ν by Deodhar’s criterion (Proposition 1.8(c)) 

Remark 2.6 (a) Theorem 2.3 shows that the tightness of a one-sided quotient is

a purely order-theoretic property, and thus depends only on the generating set J for the stabilizer of θ In other words, we may define (W J , <B) to be tight if and only if

(W θ, <B) is tight for some (equivalently, every) dominant θ ∈ V with stabilizer WJ.(b) On the other hand, the tightness of the Bruhat ordering of an orbit representation

of a double quotient is not purely a property of the underlying poset For example,

suppose W = E8 and that s, t ∈ S are the simple reflections such that Whsi ∼=D7 and

W hti ∼=A7 If θs and θt are any dominant points whose stabilizers are Whsi and Whti,

then it happens that (W θs)hti is tight and (W θt)hsi is not, even though the Bruhat

orderings of (W θs)hti and (W θt)hsi are necessarily isomorphic

Question 2.7 Is the infinite dihedral group the only (irreducible) infinite Coxeter

group W with a nontrivial quotient (W J , <B) that is tight?

Theorem 4.9 below answers this question affirmatively for the affine Weyl groups

3 Tight quotients of finite Coxeter groups.

In this section, we assume that W (and hence Φ) is finite One knows in this case that

h , i is necessarily positive definite on the span of Φ, and there is no harm in assuming

that it is positive definite on all of V , whether or not Φ spans V

It will be convenient to explicitly name the simple reflections s1, , s n and the

corresponding simple roots α1, , α n We will also modify the notation of Section 1slightly by setting hii := S − {s i }.

We let ω1, , ω ndenote the fundamental weights; i.e., the basis of Span Φ dual to thesimple co-roots Thushω i , α ∨ j i = δ ij for all i and j, ωi is dominant, and the W -stabilizer

of ωi is the parabolic subgroup Whii

A Orbit stratification and the double weight arrangement.

A useful necessary condition for (W θ, <B) to be tight involves the stratification ofsub-orbits generated by parabolic subgroups More specifically, the parabolic subgroup

W hii fixes ωi, and hence the functional µ 7→ hµ, ωi i (i.e., the coefficient of α ∨

i in µ)

is constant on Whii-orbits If this functional separates the Whii-orbits in W θ (i.e.,

µ, ν ∈ W θ are in the same W hii-orbit if and only if the coefficients of α ∨ i in µ and ν are the same), then we say that the Whii-orbits in W θ are stratified.

Proposition 3.1 If (W θ, <B) is tight, then the Whii -orbits in W θ are stratified Proof Suppose we have µ, ν ∈ W θ such that the coefficient of α ∨ i in µ − ν is zero Without loss of generality, we may assume that µ and ν are dominant relative to Φhii;

i.e., µ, ν ∈ (W θ)hii Given that (W θ, <B) is tight, we know that the Bruhat ordering of

(W θ)hii must be a chain (Theorem 2.3), and hence µ and ν must be related; say µ 6Bν.

However, each Bruhat-increasing application of a reflection subtracts a positive multiple

of a positive root, and µ − ν ∈ Span Φhii, so this is possible only if every chain from

µ to ν is generated by reflections corresponding to roots in Φ hii In turn, this forces µ and ν into the same Whii-orbit, and hence µ = ν 

Remark 3.2 (a) It is not hard to see that the above property cannot characterize

tightness For example, the Whii-orbits in a sufficiently generic W -orbit are always ified, but according to the classification below, the Bruhat ordering of a generic W -orbit

strat-is tight only if W strat-is of rank 6 2 On the other hand, it turns out that a slightly stronger

form of orbit stratification does characterize tightness—see Corollary 3.10 below

(b) Since Whii-orbits in W θ are in bijection with double cosets (Proposition 1.5(b)),

it follows that stratification of Whii-orbits may be tested by verifying that the number

of distinct values for hµ, ω i i, as µ varies over W θ, equals the number of double cosets

W hii \W/W J , where WJ denotes the stabilizer of θ In turn, one may count these double cosets by computing the character inner product for the permutation actions of W on the orbits of ωi and θ In practice, it is usually easier to explicitly generate (W θ)hii bythe algorithm suggested in Remark 1.6(a), and test stratification directly

Once the tight quotients (W hji , <B) have been classified, we claim that the tightness

of (W J , <B) may be reduced to a geometric question relating the J-th face of the

dominant chamber to the faces2of the hyperplane arrangementH = H(Φ) whose normal

vectors are the nonzero vectors of the form

xω i − yω i (x, y ∈ W, 1 6 i 6 n).

We refer to this as the double weight arrangement associated to Φ.

Lemma 3.3 The Bruhat ordering of W J is tight if and only if

(i) (W ωj , <B) is tight for all sj ∈ S − J, and

(ii) there is a face of H that contains every dominant θ ∈ V with stabilizer W J

2By a face of the hyperplane arrangement{H1, , H l }, we mean a non-empty intersection of the form

H1(ε1)∩ · · · ∩ H l (ε l ), where ε i ∈ {−, 0, +}, H(0) = H, and H(+) and H(−) denote the two half spaces

in the complement of H.

Proof Given that (W J , <B) is tight, it follows via Theorem 2.3 that (hii W J , <B) is

a chain, say w1 <B w2 <B · · · <Bw m Hence w1θ  w2θ  · · ·  w m θ for all dominant

θ with stabilizer W J, and therefore

hw1θ, ω i i > hw2θ, ω i i > · · · > hw m θ, ω i i.

We must have strict inequality here, or else the Whii-orbits in W θ would fail to stratify,

contradicting Proposition 3.1 Now since the value ofhwθ, ω i i = hθ, w −1 ω i i is unchanged

if we replace w with any member of the double coset Whii wW J , it follows that if x −1 and y −1 are in the same double cosets as wk and wl (respectively), then

hθ, xω i i > hθ, yω i i ⇔ k 6 l,

and equality occurs if and only if k = l Thus θ is confined to a single face of H Also, for every sj ∈ S − J, we have W hji ⊆ W J, so (hii W hji , <B) is a subposet of(hii W J , <B); i.e., a chain Hence (W ωj , <B) is tight, by Theorem 2.3

Conversely, given that (i) and (ii) hold, choose any dominant θ ∈ V with stabilizer

W J and suppose there exist µ, ν ∈ W θ with µ < ν but µ 66B ν We have µ = xθ

and ν = yθ for some (unique) x, y ∈ W J , hence x 66B y (Proposition 1.1(b)), and thus

xω j 66Byω j for some j such that sj ∈ S − J (Corollary 1.4) Given the hypothesis that

(W ωj , <B) is tight, it follows that xωj 6< yω j, and hence hxω j − yω j , ω i i < 0 for some i.

Now let θ(t) := tθ + (1 − t)ωj (06 t 6 1) parameterize the line segment from ωj to θ.

It is easy to see that θ(t) is dominant and has stabilizer WJ for t > 0 However,

hθ(t), x −1 ω i − y −1 ω i i = hxθ(t) − yθ(t), ω i i

is a linear function of t that is < 0 for t = 0 and > 0 for t = 1 (since µ < ν), so there must be some t > 0 such that θ(t) lies on the negative side of the hyperplane orthogonal

to x −1 ω i − y −1 ω i and θ does not, contradicting (ii) 

Since Coxeter diagrams are acyclic, the following result shows (via Lemma 3.3) that

if (W J , <B) is tight and W is irreducible, then W J has co-rank at most 2 (i.e.,|J c | 6 2).

Lemma 3.4 If W is irreducible and there is a face of H that contains all dominant

θ with stabilizer W J , then every pair s i , s j ∈ S − J is adjacent in the Coxeter diagram, and if the edge between i and j is simple (i.e., (s i s j) = 1), then W is of type A.

Proof If W is not of type A and not a dihedral group I2(m) with m odd, then

every orbit of roots is generated by a fundamental weight or a scalar multiple thereof(depending on the normalization) For the crystallographic cases, this is equivalent tothe fact that in the extended Dynkin diagram, the “extra” node is adjacent to only one

node except in type A For the non-crystallographic groups I2(m), H3, and H4, this is

an easy calculation

Now suppose that there exist (positive) roots βi and βj in the same W -orbit such

that hω i , β i i > 0, hω j , β j i > 0, and hω i , β j i = hω j , β i i = 0 If so, then for all sufficiently

small dominant θ with stabilizer WJ, the quantity

hθ + tω i+ (1− t)ω j , β i − β j i

is negative for small t > 0 and positive for large t < 1, and thus the set of dominant vectors with stabilizer WJ is not confined to one side of the hyperplane orthogonal to

β i − β j However the preceding discussion shows that if W is of rank > 2 and not of type A, then βi −β j is one of the normal vectors for the arrangementH, a contradiction.

We may take βi = αi and βj = αj provided that αi and αj are in the same W -orbit; i.e., nodes i and j must be connected by a path in the Coxeter graph that avoids edges

of even weight This eliminates all cases except those of type A and those in which the unique path from i to j passes through a (necessarily unique) edge of weight > 3.

In the latter case, it suffices to show that i and j must be adjacent If this fails, then either i or j must fail to be an endpoint of the edge of weight > 3 (or both) Interchanging i and j if necessary, we assume that i has this property, and that k is the first node on the unique path from i to j In that case, αi and αk are in the same

W -orbit, and α k and αj are in the same irreducible component of Φhii, so we may take

β i = αi and βj to be the root in the Whii-orbit of αk that is dominant relative to Φhii

The remaining possibility is that W is of type A Although not all root differences

β i − β j are normal vectors for the double weight arrangement in type A, it happens that the normal vectors do include the differences of orthogonal roots Indeed, one may check that all such differences form a single W -orbit in {xω2− yω2 : x, y ∈ W } Thus

if i and j are not adjacent, the hyperplane orthogonal to αi − α j is in H, so we may

again reach a contradiction by choosing βi = αi and βj = αj 

B The classification.

In order to identify the tight quotients of finite Coxeter groups, we first collect someadditional necessary conditions related to orbit stratification

Lemma 3.5 If Ψ ⊆ Φ is an orbit of roots that includes two orthogonal simple roots,

then for some k, the W hki -orbits in Ψ are not stratified (and thus (Ψ, <B) is not tight).

Proof If α i and αj are in the same W -orbit, then nodes i and j in the Coxeter diagram are connected by a path Given that αi and αj are orthogonal, nodes i and j must also

be non-adjacent Now since the diagrams of finite Coxeter groups are acyclic, there must

be an intermediate node k along a path connecting i and j whose removal leaves i and

j in separate components Thus α i and αj belong to distinct irreducible components of

Φhki, and hence in distinct Whki-orbits On the other hand, the coefficients of αk in αi and αj are both zero, so the Whki-orbits in Ψ cannot be stratified 

Once a particular W -orbit is known to violate stratification, the following result

allows us to extend the violation to orbits of larger Coxeter groups

Lemma 3.6 If θ is dominant and the W hji -orbits in W θ are stratified, then the

W J∩hji -orbits in W J θ are stratified for all J such that {s j } ⊆ J ⊆ S.

Proof Suppose µ, ν ∈ W J θ and that the coefficient of α j in µ−ν is zero Without loss

of generality, we may assume that µ and ν are both dominant relative to ΦJ∩hji Since

θ is the maximum element of the standard ordering of W J θ, we have µ = θ − γ for some

γ in the nonnegative span of Φ+J Moreover, since hα i , α k i 6 0 for i 6= k, it follows that

γ is anti-dominant relative to Φ J c , and thus µ and (similarly) ν are dominant relative

A n

3 2

B n

3 2

2

E7

5 4 3

2

E8

5 4 3

Figure 2: The maximal tight quotients (rank W > 3; • = tight).

to Φhji Given that the Whji-orbits in W θ are stratified, this is possible only if µ and ν are in the same Whji-orbit; since both are Φhji-dominant, this forces µ = ν 

Lemma 3.7 If the Whii -orbits in W ω j are stratified, then the W hji -orbits in W ω i

are also stratified.

Proof Suppose µ, ν ∈ W ω i satisfyhµ, ω j i = hν, ω j i We may choose x, y ∈ W so that

µ = xω i and ν = yωi, whence

hω i , x −1 ω j i = hµ, ω j i = hν, ω j i = hω i , y −1 ω j i.

Given that the Whii-orbits in W ωj are stratified, this forces x −1 ω j and y −1 ω j to be in

the same Whii-orbit, and thus wx −1 ω j = y −1 ω j for some w ∈ Whii However, Whji is the

stabilizer of ωj , so we have ywx −1 ∈ W hji, hence yw ∈ Whji x, and therefore ν = ywω i

is in the same Whji-orbit as µ = xωi 

We are now ready for the classification of tight one-sided quotients

Theorem 3.8 If W is finite and irreducible, then (W J , <B) is tight if and only if

W is of rank at most 2, or J = S, or one of the following holds:

Proof The Bruhat ordering of every nontrivial quotient of a dihedral group is a chain,

so all such Coxeter groups and their quotients have tight Bruhat orders by Theorem 2.3

Also, we have previously noted (see Example 2.2) that if θ is minuscule with respect to Φ

or one of its renormalizations, then (W θ, <B) is tight Bearing in mind the classification

of minuscule weights (e.g., see Exercise VI.4.15 of [B]), this accounts for all of the cases

listed above with|J c | = 1 except for those involving F4, H3, and the Bn-orbit of ω2.

The case W = F4 Let Ψ denote the orbit of short roots in a finite crystallographic

root system It is well-known and easy to show that if α covers β in the standard ordering of Ψ, and β is positive, then α − β must be a simple root (e.g., Proposition 3.2

of [S3]), and hence α <B β (cf Remark 1.2(b)) Dual reasoning shows that the same

must be true if α is negative The remaining possibility for a covering relation in the standard order is that α is positive and β is negative, in which case α = αi and β = −αj for some i and j If the corresponding nodes are adjacent in the Coxeter diagram, then

α − β = α i + αj is a (positive) root, and hence αi <B −α j Thus if all pairs of nodes

corresponding to short simple roots are adjacent (in particular, this means that there

can be at most two short simple roots), then (Ψ, <B) is tight For the two cases in

question, the F4-orbits of ω1 and ω4 are both root orbits; each has exactly two simpleroots, the corresponding nodes are adjacent, and we can normalize the root system sothat either orbit is the short orbit Hence both orbits have tight Bruhat orders

The case W = H3 The Bruhat orderings of the H3-orbits of ω1 and ω3 are displayed

in Figure 3, with the covering edges generated by the i-th simple reflection labeled i.

The non-minimal vertices corresponding to Φhii-dominant points are those that are the

upper endpoint of an edge labeled i, and no other labeled edge It is easy to check that

these sets of vertices form chains, so Theorem 2.3 implies that the two orders are tight

The case W = B n , J c = {s2} In the standard realization, B n acts as the group

of signed permutations of the coordinates in V = R n, and the dominant chamber may

be taken to consist of vectors with weakly increasing, nonnegative coordinates With

these choices, the vector θ = (0, 1, , 1) is dominant with stabilizer Wh2i, and thevectors that are dominant with respect to Φhii are those µ = (a1, , a n) such that

06 a1 6 · · · 6 ai−1 and ai 6 ai+1 6 · · · 6 an Hence,

(Bn θ) hii=



{µ0, µ1, , µ n−i , ν0, ν1, , ν n−i+1 } if i > 1,

{µ0, µ1, , µ n−1 } if i = 1,

where µj = (1i−1(−1) j01n−i−j ) and νj = (01i−2(−1) j1n−j−i+1 ), using the notation a k

to denote a string of k a’s Noting that νj − µ j , µj − ν j+1, and µj − µ j+1 are eachpositive multiples of positive roots, it follows via Remark 1.2(b) that

ν0 <Bµ0 <B ν1 <B µ1 <B· · · <Bν n−i+1 (i > 1),

µ0 <Bµ1 <B · · · <B µ n−1 (i = 1).

Thus Theorem 2.3 implies that (Bn θ, <B) is tight

1 2 1

2 3 3

1

2 3 3

1 2

2 3

2 2

1

3 2 1 2 1

1

3

3

3 2 1 2

Figure 3: The Bruhat orderings of H3ω1 and H3ω3.

The remaining listed cases involve quotients by non-maximal parabolic subgroups WJ with J c = {s j , s j+1 } (for W = A n) and J c = {s1, s2} (for W = B n) By Lemma 3.3,one may deduce that these quotients are tight by verifying that the corresponding face

of the dominant chamber is confined to a single face of H We omit the proof, since a

stronger result for affine Weyl groups will be given later in the proof of Theorem 5.12.Turning to the converse, we seek to show that the Bruhat orderings of all remainingquotients are not tight For the cases with |J c | = 1, our strategy is to show that the

corresponding orbits fail the stratification test in Proposition 3.1 Once this is complete,Lemmas 3.3 and 3.4 combine to eliminate the cases with|J c | > 1.

For the former, note that the following fundamental weights are roots that include

an orthogonal pair of simple roots in their orbits: ωn−1 in Bn and Dn (n > 4 only),

ω2 in H3 and E6, ω1 in E7, ω8 in E8, and ω4 in H4 It follows via Lemma 3.5 thatthe Bruhat orderings of their orbits fail the stratification test More generally, if the

j-th fundamental weight for W generates an orbit that fails the stratification test, then

Lemma 3.6 implies that the same is true for the j-th fundamental weight for any finite Coxeter group that includes W as a parabolic subgroup In this way one may deduce that all remaining orbits W ωj fail the stratification test except possibly F4ω2, F4ω3,

H4ω1, and H4ω3.

The cases W ω j = F4ω2 and F4ω3 If we normalize the root system so that Φ is

crystallographic and α1 and α2 are short, then an easy calculation shows that

s2s1s3s2s3ω3 = ω3− 2α1− 4α2− 2α3 =: µ,

s2s3s2s4s3ω3 = ω3− 4α2− 3α3− α4 =: ν.

In particular, the coefficient of α2 in µ − ν is zero On the other hand, Wh2i is the directproduct of the subgroups generated by{s1} and {s3, s4}, so if µ and ν were in the same

W h2i-orbit, then the coefficient of α1 in either µ − ν or µ − s1ν = µ − ν + 4α1 (i.e., ±2)

would have to be zero Thus the Wh2i-orbits in W ω3 are not stratified, and hence by

symmetry (Lemma 3.7), the same is true for the Wh3i-orbits in W ω2.

The cases W ω j = H4ω1 and H4ω3 Another easy calculation reveals that

s3s2s1s2s3s1s2s1ω1 = ω1 − (2 + 2r)α1− (1 + 3r)α2− (1 + 2r)α3 =: µ,

s3s2s1s4s3s2s1ω1 = ω1 − (1 + r)α1− (1 + 2r)α2− (1 + 2r)α3− rα4 =: ν,

where r = −hα1, α ∨2i = (1+ √ 5)/2 denotes the golden ratio In particular, the coefficient

of α3 in µ − ν is zero On the other hand, Wh3i is the direct product of the subgroupsgenerated by {s1, s2} and {s4}, so if µ and ν were in the same W h3i-orbit, then thecoefficient of α4 in either µ − ν or s4µ − ν = µ − ν − (1 + 2r)α4 (i.e., r or −(1 + r)) would have to be zero Thus the Wh3i-orbits in W ω1 are not stratified, and hence by

symmetry (Lemma 3.7), the same is true for the Wh1i-orbits in W ω3. 

Bearing in mind that the W -orbit of a fundamental weight is generated by some irreducible component of W , a corollary of the above proof is that whenever (W ωj , <B)

is not tight, there is some index i such that the Whii-orbits in W ωj are not stratified.Thus we obtain the following partial converse to Proposition 3.1 It would be interesting

to have a conceptual argument for it that avoids using the classification

Theorem 3.9 A fundamental orbit (W ωj , <B) is tight if and only if the Whii -orbits

in W ω j are stratified for all i.

Combining the above result with Lemma 3.3 yields a characterization of (finite) tightquotients that involves confining a face of the dominant chamber to a face of the doubleweight arrangement However, face-confinement is also related to orbit stratification, sothis leads to yet another characterization of tightness

Corollary 3.10 The Bruhat ordering of W J is tight if and only if the W hii -orbits

in W θ are stratified for all i and all dominant θ fixed by W J

Proof If θ ∈ V is dominant and fixed by W J , then the stabilizer of θ is some parabolic subgroup WI such that J ⊆ I Hence hii W J ⊇ hii W I for all i, and it follows via Theorem 2.3 that if (W J , <B) is tight, then the same must be true for (W θ, <B) In

particular, the Whii-orbits in W θ must be stratified (Proposition 3.1).

Conversely, assume that the stated condition holds but that (W J , <B) fails to be

tight Since WJ fixes each fundamental weight ωj such that sj ∈ S − J, Theorem 3.9

implies that each orbit (W ωj , <B) must be tight It then follows via Lemma 3.3 that

the set of dominant vectors with stabilizer WJ is not confined to a single face of the

double weight arrangement Hence, there must be a hyperplane H ∈ H and a dominant pair θ, θ 0 ∈ V with stabilizer W J such that H includes θ but not θ 0 Selecting a normal

vector for H, there must be some x, y ∈ W and some i such that hθ, xωi i = hθ, yω i i,

and hence µ = x −1 θ and ν = y −1 θ are two points in W θ such that the coefficient of α ∨ i

in µ − ν is zero Given that the Whii-orbits in W θ are stratified, this forces µ and ν into the same Whii-orbit However in that case, µ 0 = x −1 θ 0 and ν 0 = y −1 θ 0 must also be in

the same Whii-orbit, and this contradicts the choice of H 

Remark 3.11 (a) One sees from the classification that the (irreducible) finite Weyl

groups W such that every quotient (W hji , <B) is tight are those of rank 6 3 and the

symmetric groups This is roughly equivalent to a result of Deodhar [D2].

(b) On a case-by-case basis (aided by machine computations), we have checked that

the Whii-orbits in W ωj are stratified if and only if ((W ωj)hii , <B) is a chain Bearing

in mind Theorem 2.3, this may be viewed as a sharper version of Theorem 3.9

(c) If WJ is the stabilizer of a minuscule weight, then the longest element wJ ∈ W J

is fully commutative in the sense of [S1] However, there exist other quotients whose

longest elements are fully commutative, such as H3h3i, and it is possible to show directly

(without the classification) that this property implies tightness Indeed, given that wJ

is fully commutative, one knows that (W J , <B) is a distributive lattice (Theorems 3.2

and 7.1 of [S1]) Furthermore, one may construct the subposet of join-irreducible

ele-ments by building the heap of any reduced expression for wJ This subposet includesthe non-identity elements ofhii W J, and there is a bijection between these elements and

the occurrences of si in the reduced expression The rules of construction show that

these elements are totally ordered in the heap (for details, see [S1]), and thus a fully

commutative quotient fits the tightness criterion of Theorem 2.3

4 Affine Weyl groups and their quotients.

Now we turn to the Coxeter groups of affine type; these are the Weyl groups of affineKac-Moody algebras As with the finite Weyl groups, there is a geometric meaningassociated with the Bruhat order—there are analogues of the flag variety for affine Kac-Moody groups, these varieties have cell decompositions indexed by the Weyl group, and

the relation of inclusion of cell closures is the Bruhat order (For details, see [K].)

In this section, Φ shall denote a finite crystallographic root system (i.e.,hα, β ∨ i ∈ Z

for all α, β ∈ Φ), and W the corresponding finite Coxeter group In this situation, the

roots and co-roots both generate lattices, denoted ZΦ and ZΦ∨, respectively As in

the previous section, we let α1, , α n denote simple roots, s1, , s nthe corresponding

simple reflections, and ω1, , ω n the fundamental weights We may also assume that

h , i is positive definite on V , the ambient space for Φ.

A A linear-affine dictionary.

The affine Weyl group fW associated to Φ is typically represented as a group of

affine transformations of V On the other hand, affine Weyl groups are also Coxeter

groups, and thus are also representable as groups of linear transformations generated

by reflections through central hyperplanes (in non-Euclidean spaces) These two points

of view will be important in our analysis, so we begin with an explicit description oftheir relationship; we have not seen this particular approach used elsewhere

Assuming for initial simplicity that Φ is irreducible, let us introduce the space

e

V := V ⊕ Rδ ⊕ Rδ 0 ,

where δ and δ 0 denote two new coordinates Extend the bilinear form to eV by defining

haδ + a 0 δ 0 + µ , bδ + b 0 δ 0 + νi = ab 0 + a 0 b + hµ, νi (µ, ν ∈ V )

for all scalars a, b, a 0 , b 0 Note that δ and δ 0 are both orthogonal to V , and that h , i is

nondegenerate on eV and positive semidefinite on V ⊕ Rδ and V ⊕ Rδ 0

We define the affine root system associated to Φ to be

eΦ := {kδ + β : k ∈ Z, β ∈ Φ} ⊂ eV.

As a set of simple roots for eΦ, we use the simple roots α1, , α n for Φ, together with

α0 := δ − e α,

where eα ∈ Φ denotes the highest root (i.e., the unique root in Φ that is maximal with

respect to the standard order) In these terms, the positive roots consist of those roots

kδ + β such that either k > 0, or k = 0 and β ∈ Φ+

The affine Weyl group fW corresponding to Φ may be defined as the subgroup of GL( e V ) generated by e S := {s0, , s n }, where s i now denotes reflection through thehyperplane in eV orthogonal to α i To show that this construction yields a legitimateCoxeter system with root system eΦ amounts to checking that (i) the angle between

α i and αj is π(1 − 1/mij), where [mij] is some Coxeter matrix, and (ii) every root

kδ + β is in the f W -orbit of some simple root The former is easy and the latter is a

straightforward induction with respect to k.

Turning to the representation of fW as a group of affine transformations of V , let us

define a linear transformation tµ : eV → e V for each µ ∈ V by setting

t µ(δ 0 ) = δ 0 − 1

2kµk2δ + µ, t µ(δ) = δ, t µ(θ) = θ − hθ, µiδ (θ ∈ V ),

wherekµk :=phµ, µi denotes the usual Euclidean norm Clearly t0 is the identity map.

Proposition 4.1 We have

(a) tµ t ν = tµ+ν for all µ, ν ∈ V

(b) skδ+β = sβ t kβ ∨ = t−kβ ∨ s β for all roots kδ + β ∈ e Φ.

(c) T (Φ ∨) :={t µ : µ ∈ ZΦ ∨ } is an abelian subgroup of f W

(d) sβ t µ = ts β (µ) s β for all β ∈ Φ and µ ∈ V

(e) fW is the semi-direct product T (Φ ∨)o W